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Class 10 Triangles: cheat sheet
CBSE Mathematics. Similar triangles have equal angles and sides in the same ratio; the Basic Proportionality Theorem and the AA, SSS and SAS tests decide when triangles are similar.
Key ideas
- Similar figures have the same shape; their sizes may differ. All circles are similar, and so are all squares.
- Two triangles are similar when their corresponding angles are equal and their corresponding sides are in the same ratio.
- Basic Proportionality Theorem (Thales): a line parallel to one side of a triangle divides the other two sides in the same ratio.
- Converse: if a line divides two sides of a triangle in the same ratio, it is parallel to the third side.
- AA: two pairs of equal angles are enough. SSS: all three pairs of sides in the same ratio. SAS: one equal angle with the sides around it in the same ratio.
- Write the similarity in matching order: △ABC ~ △PQR means A ↔ P, B ↔ Q, C ↔ R.
- Perimeters of similar triangles are in the same ratio as their sides.
Remember
| BPT (DE ∥ BC) | AD/DB = AE/EC |
| △ABC ~ △PQR | AB/PQ = BC/QR = CA/RP |
| Shadows at the same time | height₁/shadow₁ = height₂/shadow₂ |
Watch out
- Matching the wrong vertices, so the wrong sides are put in a ratio.
- Using SAS when the equal angle is not between the two proportional sides.
- Writing AD/AB = AE/EC (mixing a part with a whole).
Worked example
Isha's eyes are 1.6 m above the ground. A mirror lies flat on the ground 1 m from Isha's feet, and in it Isha sees the top of a tree that is 5 m from the mirror. How tall is the tree?
Working: the angles at the mirror are equal, so the triangles are similar: height/5 = 1.6/1
Answer: 8 m
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